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Gluing of singular and critical points. (English) Zbl 0905.14008

The author extends Viro’s method of glueing polynomials in order to keep singular or critical points in the process. The input for the glueing method is a subdivision $\left\{{{\Delta }}_{i}\right\}$ of a nondegenerate Newton polyhedron and a compatible system of polynomials ${F}_{i}$ with support on the ${{\Delta }}_{i}$. A sufficient condition for the existence of a polynomial with the same singularities as the ${F}_{i}$ is roughly that for each $i$ the equisingular locus in the space of all polynomials is smooth and transversal to the space of polynomials with the given Newton diagram and coinciding with ${F}_{i}$ for all monomials in ${{\Delta }}_{i}$.

As example plane curves with the maximal number of cusps are constructed for degree eight ($\kappa =15$) and nine ($\kappa =20$). Another application is the asymptotically complete solution to the problem of possible collections of critical points of real polynomials in two variables without critical points at infinity.

##### MSC:
 14E15 Global theory and resolution of singularities 32S15 Equisingularity (topological and analytic) 14F45 Topological properties of algebraic varieties 14B05 Singularities (algebraic geometry) 14H20 Singularities, local rings